scholarly journals Oscillation Theorems for Second-Order Half-Linear Advanced Dynamic Equations on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Shuhong Tang ◽  
Tongxing Li ◽  
Ethiraju Thandapani
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Theorems

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation on an arbitrary time scale with sup , where and . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.

Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

2011 ◽  
Vol 54 (4) ◽  
pp. 580-592 ◽  
Author(s):  
Jia Baoguo ◽  
Lynn Erbe ◽  
Allan Peterson
Keyword(s):  
Difference Equation ◽  
Dynamic Equation ◽  
Time Scale ◽  
Nonlinear Function ◽  
Second Order ◽  
Dynamic Equations ◽  
Coefficient Function ◽  
The Difference ◽  
Oscillation Theorems ◽  
Type Oscillation

AbstractConsider the second order superlinear dynamic equationwhere p ∈ C(, ℝ), is a time scale, ƒ : ℝ → ℝ is continuously differentiable and satisfies ƒ ′(x) > 0, and x ƒ (x) > 0 for x ≠ 0. Furthermore, f (x) also satisfies a superlinear condition, which includes the nonlinear function ƒ (x) = xα with α > 1, commonly known as the Emden–Fowler case. Here the coefficient function p(t) is allowed to be negative for arbitrarily large values of t. In addition to extending the result of Kiguradze for (∗) in the real case = ℝ, we obtain analogues in the difference equation and q-difference equation cases.

scholarly journals Oscillation Criteria of Second-Order Mixed Neutral Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01006
Author(s):  
L M Feng ◽  
Y G Zhao ◽  
Y L Shi ◽  
Z L Han
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Second Order ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Dynamic Equation ◽  
Delay Dynamic Equations ◽  
Oscillation Theorems

In this artical, we consider a second-order neutral dynamic equation on a time scales. A number of oscillation theorems are shown that supplement and extend some known results in the eassay.

scholarly journals Oscillation for Higher Order Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Qiuli He ◽  
Hongjian Xi ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Positive Integers

We investigate the oscillation of the following higher order dynamic equation:{an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scaleT, wheren≥2,ak(t)  (1≤k≤n)andp(t)are positive rd-continuous functions onTandα,βare the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

scholarly journals Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Weisong Chen ◽  
Zhenlai Han ◽  
Shurong Sun ◽  
Tongxing Li
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Nonlinear Dynamic Equation ◽  
Oscillation Theorems

By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

scholarly journals A telescoping principle for oscillation of the second order half-linear dynamic equations on time scales

2009 ◽  
Vol 43 (1) ◽  
pp. 243-255
Author(s):  
Jiří Vítovec
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic

Abstract . We establish the so-called “telescoping principle” for oscillation of the second order half-linear dynamic equation [r(t)Φ(x<sup>Δ</sup>)]<sup>Δ</sup> + c(t)Φ(x<sup>σ</sup>) = 0 on a time scale. This principle provides a method enabling us to construct many new oscillatory equations. Unlike previous works concerning the telescoping principle, we formulate some oscillation results under the weaker assumption r(t) ≠ 0 (instead r(t) > 0).

Oscillation criteria for quasi-linear functional dynamic equations on time scales

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Samir Saker ◽  
Said Grace
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Linear Functional ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Dynamic Equations ◽  
Gamma Delta ◽  
T Tau ◽  
Positive Integers

AbstractThis paper is concerned with oscillation of the second-order quasilinear functional dynamic equation $(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$ on a time scale $\mathbb{T}$ where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on $\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}$ and $\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

scholarly journals New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Li Gao ◽  
Quanxin Zhang ◽  
Shouhua Liu
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Generalized Riccati Transformation ◽  
Delay Dynamic Equations ◽  
Inequality Technique ◽  
Nonlinear Delay

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

scholarly journals Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1897
Author(s):  
Taher S. Hassan ◽  
Yuangong Sun ◽  
Amir Abdel Menaem
Keyword(s):  
Dynamic Equation ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Recent Literature ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Nonlinear Dynamic Equations ◽  
Type Oscillation

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

scholarly journals New Oscillation Results of Second-Order Damped Dynamic Equations withp-Laplacian on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Second Order ◽  
Dynamic Equations ◽  
Coefficient Function ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Generalized Riccati Technique

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation withp-Laplacian on time scales of the form(rtφγ(xΔ(t)))Δ+ptφγ(xΔ(t))+f(t,x(g(t)))=0, where the coefficient functionp(t)may change sign. Two examples are given to demonstrate the obtained results.

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